On the Uniform Perfectness of Groups[4Pt]

ARCHIVUM MATHEMATICUM (BRNO) Tomus 55 (2019), 333–339

ON THE UNIFORM PERFECTNESS OF GROUPS OF BUNDLE HOMEOMORPHISMS

Tomasz Rybicki

Abstract. Groups of homeomorphisms related to locally trivial bundles are studied. It is shown that these groups are perfect. Moreover if the homeomorphism isotopy group of the base is bounded then the bundle homeomorphism group of the total space is uniformly perfect.

1. Introduction Let M be a topological metrizable (and so paracompact) and second countable manifold, let Homeo(M) denote the group of all homeomorphisms of M and let Homeoc(M) be the subgroup of all compactly supported elements. Next the symbol H(M) (resp. Hc(M)) stands for the subgroup of all elements of Homeo(M) that can be joined to the identity by an isotopy (resp. a compactly supported isotopy) in Homeo(M). Recall that a group G is called perfect if it is equal to its commutator subgroup, i.e. any element of G can be expressed as a product of commutators [f, g] = fgf −1g−1, where f, g ∈ G. Then we have the following theorem which essentially follows from the results of Mather [10], and Edwards and Kirby [3]. Theorem 1.1 ([7, Theorem 1.1, Cor. 1.3]). Assume that either M is compact (possibly with boundary), or M is open and admits a compact exhaustion. Then the group Hc(M) is perfect. Moreover, for M connected, Hc(M) is simple if and only if ∂M = ∅. As a topological group, Homeo(M) will be endowed with the Whitney (or graph) topology. If M is compact this topology coincides with the compact-open topology. From now on we will assume that that M, B and F are topological metrizable and second countable manifolds and π : M → B is a locally trivial bundle with the standard fiber F . Denote by Homeo(M, π) (resp. Homeoc(M, π)) the totality of fiber preserving homeomorphism (resp. with compact support). Next denote by Homeoπ(M) the group of all bundle homeomorphisms of π. It is clear that π induces the homomorphism P : Homeoπ(M) → Homeo(B) given by P (f)(π(x)) = π(f(x)).

2010 Mathematics Subject Classification: primary 57S05; secondary 58D05, 55R10. Key words and phrases: homeomorphism group, uniformly perfect, continuously perfect, bounded, locally trivial bundle. Partially supported by the local grant n. 16.16.420.054. Received March 23, 2019. Editor M. Čadek. DOI: 10.5817/AM2019-5-333 334 T. RYBICKI

Let Homeoπ,c(M) be the subgroup of Homeoπ(M) of all transversely compactly supported bundle homeomorphisms of M. That is, f ∈ Homeoπ,c(M) if and only if f sends each fiber onto another fiber and π(supp(f)) is compact. Then Homeoc(M, π) is a normal subgroup of Homeoπ,c(M). The symbol Hπ,c(M) (resp. Hc(M, π)) stands for the group of all homeomorphisms from Homeoπ,c(M) (resp. Homeoc(M, π)) that can be joined to the identity by an isotopy in Homeoπ,c(M) (resp. Homeoc(M, π)). It follows the existence of the homomorphism P : Hπ,c(M) → Hc(M, π).

Theorem 1.2. Let π : M → B be a locally trivial bundle with the standard fiber F . If F is closed or is the interior of a compact manifold with boundary then the group Hc(M, π) is perfect. Moreover, if F is closed then Hπ,c(M) is perfect too.

Clearly these groups are not simple. For a topological group G by PG we denote the isotopy or path group of G, that is the totality of continuous paths f : I → G with f(0) = e, I = [0, 1]. A subgroup G ≤ Homeo(M) is called fragmentable if each element of G can be written as a product of homeomorphisms from G supported in open balls. Next G is said to be path fragmentable if the path group PG is fragmentable. Observe that the group Hc(M) is path fragmentable (and so fragmentable) due to Theorem 3.3 below. Recall that a group is called bounded if it is bounded with respect to any bi-invariant metric on it. Next a group G is uniformly perfect if any element can be expressed as a product of a bounded number of commutators. Clearly any bounded and perfect group is uniformly perfect. See Section 2 for more details. The main result is the following

Theorem 1.3. Assume that π : M → B is a locally trivial bundle with the standard fiber F closed. Then we have:

(1) If Hπ,c(M) is uniformly perfect then Hc(B) is also uniformly perfect.

(2) If the fragmentation norm on the group PHc(B) is bounded, then Hπ,c(M) is uniformly perfect and

cldHπ,c(M) ≤ fdPHc(B) + 2(n + 1) ,

where n = dim B and cld (resp. fd) is the commutator length diameter (resp. fragmentation diameter), cf. Section 2.

The problem of the algebraic structure of homeomorphism groups, especially the boundedness and uniform perfectness of them, has drawn much attention. It has been studied among others in [1], [4], [5], [6], [7], [8], [9], [11], [12], [14], [15], [16] (see also references therein). Throughout all manifolds are topological, second countable and metrizable. The symbol of composition in homeomorphism groups will be omitted. GROUPS OF BUNDLE HOMEOMORPHISMS 335

2. Conjugation-invariant norms The notion of boundedness can be expressed in terms of a conjugation-invariant norms. A conjugation-invariant norm on a group G is a function ν : G → [0, ∞) which satisfies the following conditions. For any g, h ∈ G (1) ν(g) > 0 if and only if g 6= e, (2) ν(g−1) = ν(g), (3) ν(gh) ≤ ν(g) + ν(h), (4) ν(hgh−1) = ν(g). It is easily seen that G is bounded if and only if any conjugation-invariant norm on G is bounded. By the diameter of ν we mean supg∈G ν(g). Let G be a group and let [G, G] be its commutator subgroup. For g ∈ [G, G] the symbol clG(g) stands for the least k such that g is written as a product of k commutators and is called the commutator length of g. Observe that the commutator length clG is a conjugation-invariant norm on [G, G]. In particular, if G is a perfect group then clG is a conjugation-invariant norm on G. For any perfect group G denote by cldG the commutator length diameter of G, cldG := sup clG(g). g∈G Then G is uniformly perfect iff cldG < ∞. Another example of conjugation-invariant norm is the following. Let G be a subgroup of Homeo(M) and assume that G is fragmentable. For h ∈ G, h 6= id, we define the fragmentation norm fragG(h) to be the smallest integer r > 0 such that h = h1 . . . hr with supp(hi) ⊂ Ui for all i, where Ui is a ball. By definition fragG(id) = 0. Next by fdG we denote the fragmentation diameter of G, fdG := suph∈G fragG(h). Recall the notion of displacement of a subgroup. A subgroup K of G is called strongly m-displaceable if there is g ∈ G such that the subgroups K, gKg−1,..., gmKg−mpairwise commute. Then we say that g m-displaces K. Proposition 2.1 ([2]). Assume that g m-displaces K ≤ G for every m ≥ 1. Then for any f ∈ [K,K] we have clG(f) ≤ 2.

3. Locally continuously fragmentable groups The following type of fragmentations is important when studying groups of homeomorphisms. Definition 3.1. A subgroup G ≤ Homeo(M) is called locally continuously frag- d mentable with respect to a finite open cover {Ui}i=1 if there exist a neighborhood N of id ∈ G and continuous mappings σi : N → G, i = 1, . . . , d, such that σi(id) = id and for all f ∈ N one has

f = σ1(f) ··· σd(f) , supp(σi(f)) ⊂ Ui, ∀i .

Moreover, we assume that each supp(σi(f)) is compact whenever f ∈ Homeoc(M). If N = G then G is called globally continuously fragmentable. 336 T. RYBICKI

Clearly, if G is connected and locally continuously fragmentable then it is fragmentable. Observe that Def. 3.1 can also be formulated for PG rather than G, where G ≤ H(M). For a manifold X and a subgroup G ≤ H(M), let C(X, G) stand for the group of all continuous maps X → G with the pointwise multiplication and the compact-open S x x topology. For f ∈ C(X, G) we define supp(f) = x∈X supp(f ), where f : p ∈ M 7→ f(x)(p) ∈ M. Then Def. 3.1 extends obviously for C(X, G). It is easy to check that if G is a topological group then C(X, G) is also a topological group. Proposition 3.2. If X is compact and G is locally continuously fragmentable with d respect to {Ui}i=1, then so is C(X, G).

Proof. Let σi : N → G, i = 1, . . . , d, be as in Def. 3.1. Define Nˆ = {f ∈ C ˆ C(X, G): f(X) ⊂ N } and continuous maps σi : N → C(X, G) by the formulae C x C x σi (f)(x) = σi(f ), where f ∈ C(X, G), x ∈ X. It follows that supp(σi (f) ) ⊂ Ui C for all i and x. Consequently we have supp(σi (f)) ⊂ Ui. The results of this paper depend essentially on the deformation properties for the spaces of imbeddings obtained by Edwards and Kirby in [3]. See also Siebenmann [17]. All manifolds are assumed to be metrizable and second countable (i.e. have at most countably many connected components). A ball U in an n-dimensional manifold M is a relatively compact, open ball imbedded with its closure in M. For M closed, let d = dM is the smallest integer Sd such that M = i=1 Ui where Ui is a ball if M is connected (or Ui is the union of disjoint open balls, each ball lies in a different connected component, if M is not connected) such that cl(Ui) 6= M for each i. Then d ≤ n + 1 . Next if M is an open n+1 manifold, then M admits an open cover {Ui}i=1 (see [13]), called a Palais cover, such that each Ui is the union of a countable, locally finite family of balls with pairwise disjoint closures. In each case such a cover will be called related to M. Theorem 3.3. Assume that M is closed or M is the interior of a compact manifold. Then Hc(M) is a locally continuously fragmentable group (Def. 3.1) with respect to d any cover {Ui}i=1, d ≤ n + 1, related to M. Moreover the isotopy group PHc(M) d is also locally continuously fragmentable with respect to {Ui}i=1. For the proof see Prop. 3.2 [16] and the proof of Theorem 1.3 in [16]. The second claim follows from Prop. 3.2 above.

Proposition 3.4. If π : M → B be a locally trivial bundle then PHc(M, π) is −1 globally continuously fragmentable with respect to {π (Ui)}, where {Ui} is a cover related to B. Next if F is closed or is the interior of a compact manifold with boundary, and {Vj} is a cover related to F , then PHc(M, π) is locally continuously fragmentable with respect to {Ui × Vj)}i,j. d Proof. To show the first claim, suppose that {λi}i=1 is a partition of unity d t subordinate to {Ui}i=1, a cover related to B. Let g = {g } ∈ PHc(M, π). t λ1(π(p))t For p ∈ M put h1(p) = g (p). Next

t λ1(π(p)t) −1 (λ1+λ2)(π(p))t h2(p) = (g ) g (p) . GROUPS OF BUNDLE HOMEOMORPHISMS 337

In general, for 3 ≤ i ≤ d we define

t (λ1+···λi−1)(π(p))t) −1 (λ1+···+λi)(π(p))t hi(p) = (g ) g (p) . t −1 t t t Then supp(hi) ⊂ π (Ui) for all i and t, and g = h1 . . . hd. Moreover, the maps t σi : g 7→ hi = {hi} are continuous. d In order to show the second assertion let g ∈ PHc(M, π) and {Ui}i=1 be related to B so that π trivializes over each Ui. We suppose that g is so small that for t t each hi as above we have that hi|π−1(x) ∈ N , where N is as in Def. 3.1, for all t x ∈ Ui, i = 1, . . . , d. Then we apply Theorem 3.3 to the family hi|π(x), x ∈ Ui, in a fiberwise fashion, with respect to a cover {Ui × Vj}j, where {Vj} is related to F . It follows that PHc(M, π) is locally continuously fragmentable with respect to {Ui × Vj}i,j. The following basic lemma for homeomorphisms is no longer true in the C1 category. Lemma 3.5 ([10], [16]). Let W and V be balls such that cl(W ) ⊂ V . Then there exist φ ∈ Hc(V ) and a continuous map S : Hc(W ) → Hc(V ) such that g = [S(g), φ] for all g ∈ Hc(W ). We also need the following Proposition 3.6. If π : M → B is a locally trivial bundle with the standard fiber F closed, then the homomorphism P : Hπ,c(M) → Hc(B) is surjective. Furthermore, the induced map for isotopies P : PHπ,c(M) → PHc(B) is also surjective. Proof. Obviously, the second assertion implies the first. To show the second, let h ∈ PHc(B). From Theorem 3.3 it follows that h = h1 . . . hp with all hi ∈ PHc(B) supported balls. Consequently, each hi can be lifted by means of a trivialization ˜ ˜ ˜ of π to an isotopy hi ∈ PHπ,c(M) such that P (hi) = hi. Thus P (h) = h, where ˜ ˜ ˜ h = h1 ... hp.

Proof of Theorem 1.2. First we prove that Hc(M, π) is perfect. According to Prop. 3.4 it suffices to assume that M = U × V , where U (resp. V ) is a ball in B (resp. in F ) and π is the projection U × V → U. If f ∈ Hc(U × V, π) then supp(f) ⊂ U × W such that cl(W ) ⊂ V . In view of Lemma 3.5 there are φ ∈ Hc(V ) and a continuous map S : Hc(W ) → Hc(V ) such that g = [S(g), φ] for all g ∈ Hc(W ). ˜ ˜ Consider S : Hc(U × W ) → Hc(U × V ) given by S(g˜)(x, y) = (x, S(g˜|π−1(x))(y)) for all g˜ ∈ Hc(U × W ). Then we get f = [S˜(f), id ×φ]. It remains to modify id ×φ ¯1 ¯ to be compactly supported. Namely, if φ = φ with φ ∈ PHc(V ) and λ: B → [0, 1] is a compactly supported bump function satisfying λ|π(supp(f)) = 1, then we may use φˆ given by φˆ(x, y) = (x, φ¯λ(x)(y)) instead of id ×φ. ¯ To show that Hπ,c(M) is perfect, let f ∈ Hπ,c(M). Take f ∈ PHπ,c(M) such that f¯1 = f. According to Theorem 3.3 we have a decomposition of isotopies ¯ P (f) = h1 . . . hr, where each hi is supported in a ball in B, say Ui (or a union of locally finite family of balls with pairwise disjoint closures). In view of Prop.3.6 ¯ ¯ ¯ ¯ ¯ there exist lifts h1,..., hr ∈ PHπ,c(M) of h1, . . . , hr, resp. If h := h1 ... hr then 338 T. RYBICKI

¯¯−1 1 g¯ = fh lies in ker(P ). Consequently, g¯ belongs to Hc(M, π) and so is a product ¯1 of commutators due to the first part. On the other hand, each hi can be viewed as an element of Hc(Ui) if we use a local trivialization over Ui. Therefore, due ¯1 ¯1 1¯1 to Lemma 3.5, each hi is a commutator. Thus f = f = g¯ h is a product of commutators and Hπ,c(M) is perfect.

4. Proof of Theorem 1.3 (1) It follows from Prop. 3.6. ¯ ¯1 (2) Let f ∈ Hπ,c(M) and let f ∈ PHπ,c(M) such that f = f. We have ¯ P (f) ∈ PHc(B) and we may use Theorem 3.3 to get a fragmentation of isotopies ¯ P (f) = h1 . . . hp , where each isotopy hi ∈ PHc(Ui), where Ui is a ball for all i, and where p is bounded ¯ −1 according to the assumption. In view of Prop. 3.6 we define hi ∈ PHπ,c(π (Ui)), the lifts of hi, and we put ¯ ¯ ¯ ¯¯−1 h = h1 ... hp and g¯ = fh . Since P (h¯) = P (f¯), it follows that g¯ is an isotopy in ker(P ) joining g = g¯1 to the identity. ¯ By using local trivialization of π, each hi can be regarded as an element of ¯1 PHc(Ui). Then due to Lemma 3.5 each hi can be written as a commutator. Now in view of Prop. 3.4 g = g¯1 can be written as a product of at most n + 1 factors, −1 g = g1 . . . gn+1 , supp(gi) ⊂ π (Ui) for all i, where Ui is a ball provided B is closed. If B is open then Ui is a finite union of balls with disjoint closures. In both cases each gi can be expressed as a product of at most two commutators. In fact, for every Ui there exists φi ∈ Hπ,c(M) such k −1 that the family {φi (π (Ui))}k∈N0 is pairwise disjoint, and we apply Theorem 1.2 and Prop. 2.1. Observe that the reasoning from [10] does not apply in this case. Consequently, f can be expressed as a product of p + 2(n + 1) commutators. Therefore Hπ,c(M) is uniformly perfect and

cldHπ,c(M) ≤ fdPHc(B) + 2(n + 1) , as required. Acknowledgement. I would like to thank very much the referee for pointing out some mistakes and unclear statements in the previous version of this note and for other valuable comments.

References [1] Anderson, R.D., On homeomorphisms as products of a given homeomorphism and its inverse, Topology of 3-manifolds (M. Fort ed., ed.), Prentice-Hall, 1961, pp. 231–237. [2] Burago, D., Ivanov, S., Polterovich, L., Conjugation invariant norms on groups of geometric origin, Adv. Stud. Pure Math. 52 (2008), 221–250, Groups of diffeomorphisms. GROUPS OF BUNDLE HOMEOMORPHISMS 339

[3] Edwards, R.D., Kirby, R.C., Deformations of spaces of imbeddings, Ann. Math. (2) 93 (1971), 63–88. [4] Epstein, D.B.A., The simplicity of certain groups of homeomorphisms, Compositio Math. 22 (2) (1970), 165–173. [5] Fisher, G.M., On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc. 97 (1960), 193–212. [6] Fukui, K., Imanishi, H., On commutators of foliation preserving homeomorphisms, J. Math. Soc. Japan 51 (1) (1999), 227–236. [7] Kowalik, A., Rybicki, T., On the homeomorphism groups on manifolds and their universal coverings, Cent. Eur. J. Math. 9 (2011), 1217–1231. [8] Lech, J., Michalik, I., Rybicki, T., On the boundedness of equivariant homeomorphism groups, Opuscula Math. 38 (2018), 395–408. [9] Ling, W., Factorizable groups of homeomorphisms, Compositio Math. 51 (1) (1984), 41–51. [10] Mather, J.N., The vanishing of the homology of certain groups of homeomorphisms, Topology 10 (1971), 297–298. [11] McDuff, D., The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold, J. London Math. Soc. (2) 18 (1978), 353–364. [12] Michalik, I., Rybicki, T., On the structure of the commutator subgroup of certain homeomorphism groups, Topology Appl. 158 (2011), 1314–1324. [13] Munkres, J., Elementary differential topology, Ann. Math. Studies, vol. 54, Princeton Uni- versity Press, 1966. [14] Rybicki, T., On commutators of equivariant homeomorphisms, Topology Appl. 154 (2007), 1561–1564. [15] Rybicki, T., Boundedness of certain automorphism groups of an open manifold, Geom. Dedicata 151 (1) (2011), 175–186. [16] Rybicki, T., Locally continuously perfect groups of homeomorphisms, Ann. Global Anal. Geom. 40 (2011), 191–202. [17] Siebenmann, L.C., Deformation of homeomorphisms on stratified sets, 1, 2, Comment. Math. Helv. 47 (1972), 123–163.

AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland E-mail: [email protected]